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Suppose individual event hazards have the additive form $h_i=h_{0i}+h_1$, where $h_{0i}$ is known and $h_1$ is common. The aggregate counting-process intensity in survival analysis is $\lambda ={\sum }_i{Y}_i{h}_{0i}+Y{h}_1$, with $Y={\sum }_i{Y}_i$ the current risk set size. Writing $dN=\lambda dt+dM$ and dividing by $Y$ givesThe last term is a counting-process martingale integral. This establishes the cumulative common-hazard target on the observed at-risk time range. The known offset must be averaged over the current risk set, not the initial sample. Between event jumps, its subtraction can make the unconstrained estimate decrease. If all known hazards equal $h_0$ and $Y>0$ up to $t$, the estimate reduces to the Nelson–Aalen estimator minus $H_0(t)$.

For individual $i$, let $N_i(t)$ count its observed event by time $t$ and let $Y_i(t)$ indicate membership of the risk set immediately before $t$. Set $N={\sum }_i{N}_i$ and $Y={\sum }_i{Y}_i$. Under a common hazard function $h(t)$ and independent censoring, a currently at-risk subject has event probability $h(t)dt+o(dt)$ in the next short interval. Summing over the current risk set gives the conditional intensityEquivalently, the event count decomposes aswhere $M$ is the mean-zero counting-process martingale obtained by subtracting its conditional compensator. Estimating the unknown hazard increment by the observed event increment divided by the risk set size gives the Nelson–Aalen estimatorHere $a_j$ are the distinct observed event times and $r_j=Y(a_j)$ includes the subject who is about to fail. A censored observation affects subsequent risk sets, but causes no event jump. This is the requested no-ties estimator; tied event counts would produce $d_j/r_j$.

The derivation also identifies the estimation error:Thus it estimates $H(t)$ on the time range where individuals remain at risk. It is not exactly unbiased for the entire cumulative hazard after the last risk set disappears. Its usual Nelson–Aalen variance estimator is ${\sum }_{a_j\le t}{r}_j^{-2}$, since the compensator variance increment of the no-ties event process is $YdH$.